Hello, Fullmetal_Hye!
Solve for \(\displaystyle x,\;\;-2\pi \,\leq x\,\leq\,2\pi\)
\(\displaystyle 1)\;\sin x\cos x \:=\:\frac{1}{4}\)
\(\displaystyle 2\sin x\cos x \:=\:\frac{1}{2}\)
\(\displaystyle \sin2x \:=\:\frac{1}{2}\)
\(\displaystyle 2x\:=\:\frac{\pi}{6},\;\frac{5\pi}{6},\;\frac{13\pi}{6},\;\frac{17\pi}{6}\)
Therefore: \(\displaystyle x \:=\:\frac{\pi}{12},\;\frac{5\pi}{12},\;\frac{13\pi}{12},\;\frac{17\pi}{12}\)
\(\displaystyle 2)\;\cos4x \:=\:\frac{\sqrt{3}}{2}\)
\(\displaystyle 4x\:=\:\frac{\pi}{6},\;\frac{11\pi}{6},\;\frac{13\pi}{6},\;\frac{23\pi}{6},\;\frac{25\pi}{6},\;\frac{35\pi}{6},\;\frac{37\pi}{6},\;\frac{47\pi}{6}\)
Then: \(\displaystyle x\:=\:\frac{\pi}{24},\;\frac{11\pi}{24},\;\frac{13\pi}{24},\;\frac{23\pi}{24},\;\frac{25\pi}{24},\;\frac{35\pi}{24},\;\frac{37\pi}{24},\;\frac{47\pi}{24}\)
I don't very well know which quadrants I should consider.
I understand your problem: you don't know "how far to go".
If we had, for example, \(\displaystyle \sin x\,=\,\frac{1}{2}\), we know that: \(\displaystyle x\:=\:\frac{\pi}{6},\;\frac{\5\pi}{6}\)
\(\displaystyle \;\;\)That is, we know there are two answers on the interval \(\displaystyle [0,2\pi]\)
If we had: \(\displaystyle \sin\)2\(\displaystyle x\,=\,\frac{1}{2}\), we would "go around twice".
\(\displaystyle \;\;\)We'd have: \(\displaystyle \,2x\:=\:\frac{\pi}{6},\;\frac{5\pi}{6}\) and \(\displaystyle \frac{13\pi}{6},\;\frac{17\pi}{6}\)
(We get the last two by adding \(\displaystyle 2\pi\) to the first two.)
Solve for \(\displaystyle x\) (divide by 2): \(\displaystyle \,x\;=\;\frac{\pi}{12},\;\frac{5\pi}{12},\;\frac{13\pi}{12},\;\frac{17\pi}{12}\)
(Note that the last one is still less than \(\displaystyle 2\pi\).)
If we have: \(\displaystyle \sin\)3\(\displaystyle x\,=\,\frac{1}{2}\), we would "go around three times:.
\(\displaystyle \;\;\)We'd have: \(\displaystyle \,\frac{\pi}{6},\;\frac{5\pi}{6}\) and \(\displaystyle \frac{13\pi}{6},\;\frac{17\pi}{6}\) and \(\displaystyle \frac{25\pi}{6},\;\frac{29\pi}{6}\)
Solve for \(\displaystyle x\) (divide by 3): \(\displaystyle \,x\:=\:\frac{\pi}{18},\;\frac{5\pi}{18},\;\frac{13\pi}{18},\;\frac{17\pi}{18},\;\frac{25\pi}{18},\;\frac{29\pi}{18}\)
See how it works?
